Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/386

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SOMMATION

12. On sait que

$e^{a}=1+{\frac {a}{1}}+{\frac {a^{2}}{1.2}}+{\frac {a^{3}}{1.2.3}}+{\frac {a^{4}}{1.2.3.4}}+\ldots \,;$

d’où

$e^{a}-1={\frac {a}{1}}+{\frac {a^{2}}{1.2}}+{\frac {a^{3}}{1.2.3}}+{\frac {a^{4}}{1.2.3.4}}+\ldots \,;$

de sorte qu’on a ici

$\operatorname {f} [a]=e^{a}-1\,;$

donc, 1.^o la somme de la série

${\frac {a\operatorname {Cos} .t}{1}}+{\frac {a^{2}\operatorname {Cos} .2t}{1.2}}+{\frac {a^{3}\operatorname {Cos} .3t}{1.2.3}}+{\frac {a^{4}\operatorname {Cos} .4t}{1.2.3.4}}+\ldots$

sera (Remarque I.)

${\frac {\left\{e^{a\left(\operatorname {Cos} .t+{\sqrt {-1}}\operatorname {Sin} .t\right)}-1\right\}+\left\{e^{a\left(\operatorname {Cos} .t-{\sqrt {-1}}\operatorname {Sin} .t\right)}-1\right\}}{2}}$

ou bien

$e^{a\operatorname {Cos} .t}.{\frac {e^{+{\sqrt {-1}}a\operatorname {Sin} .t}+e^{-{\sqrt {-1}}a\operatorname {Sin} .t}}{2}}=e^{a\operatorname {Cos} .t}.\operatorname {Cos} .(a\operatorname {Sin} .t)-1.$

de sorte qu’en ajoutant $1$ de part et d’autre, on a

$e^{a\operatorname {Cos} .t}.\operatorname {Cos} .(a\operatorname {Sin} .t)=$

$1+{\frac {a\operatorname {Cos} .t}{1}}+{\frac {a^{2}\operatorname {Cos} .2t}{1.2}}+{\frac {a^{3}\operatorname {Cos} .3t}{1.2.3}}+{\frac {a^{4}\operatorname {Cos} .4t}{1.2.3.4}}+\ldots$

2.^o La somme de la série

${\frac {a\operatorname {Sin} .t}{1}}+{\frac {a^{2}\operatorname {Sin} .2t}{1.2}}+{\frac {a^{3}\operatorname {Sin} .3t}{1.2.3}}+{\frac {a^{4}\operatorname {Sin} .4t}{1.2.3.4}}+\ldots$

sera (Remarque II)